In mathematics, the restricted product is a construction in the theory of topological groups.
Let I be an index set; S a finite subset of I. If G_i is a locally compact group for each i \in I, and K_i \subset G_i is an open compact subgroup for each i \in I \setminus S, then the restricted product
\prod_i\nolimits' G_i\,
is the subset of the product of the G_i's consisting of all elements (g_i)_{i \in I} such that g_i \in K_i for all but finitely many i \in I \setminus S.
This group is given the topology whose basis of open sets are those of the form
\prod_i A_i\,,
where A_i is open in G_i and A_i = K_i for all but finitely many i.
One can easily prove that the restricted product is itself a locally compact group. The best known example of this construction is that of the adele ring and idele group of a global field.
See also
References
- Fröhlich, A. & Cassels, J. W. (1967), Algebraic number theory, Boston, MA: Academic Press, ISBN 978-0-12-163251-9